A Neutrosophic Control Chart for Monitoring Processes with

Indeterminate Measurements: An Information-Fusion Approach

to Statistical Quality Control

Maha Ibrahim1,* Sajid Khan2

1 Faculty of Joint Degree, Tashkent State University of Economics, Uzbekistan

2 Department of Computer Science, Sukkur IBA University, Pakistan

Received: August 30, 2025 Accepted: November 28, 2025 ⋆ Corresponding author

ABSTRACT

Classical Shewhart control charts assume each measurement is a single determinate number. In many real processes—

automated gauges reporting tolerance bands, human inspectors giving ranges, or sensors whose readings are only

trustworthy within an interval—each observation is better described by a lower and an upper value together with

a degree of indeterminacy. We formulate process monitoring in the neutrosophic statistics framework, where a

measurement is a neutrosophic number xN = xL+xUIN with indeterminacy interval IN ∈ [IL, IU], and we treat the

reconciliation of the determinate and indeterminate parts as an information-fusion step. We derive neutrosophic

control limits for the process mean, propose a three-state signalling rule (in-control / watch / out-of-control), and

study the average run length (ARL) by simulation. The neutrosophic chart reduces to the Shewhart chart when

indeterminacy vanishes, raises the in-control ARL modestly, and under moderate measurement indeterminacy detects

a one-sigma mean shift with a smaller out-of-control ARL than a Shewhart chart applied to interval midpoints.

Keywords: Neutrosophic statistics Information fusion Statistical process control Control chart Average run length

Indeterminacy

1. INTRODUCTION

Statistical process control (SPC) rests on the idea that a stable

process produces measurements varying only through common

causes, and that a chart can flag the assignable causes.

The theory is built for crisp data. Yet a growing share of industrial

measurement is not crisp: vision systems report a pass

band, coordinate-measuring machines quote an uncertainty,

and manual inspection yields “about 12.3 to 12.6.” Forcing

such observations to a single number discards information

and, worse, hides how much of the observed variation is real

versus measurement indeterminacy.

The usual industrial responses to non-crisp data are unsatisfying.

One is to collapse each reading to a midpoint and

proceed with a classical chart; this throws away the width,

which is exactly the part that records how trustworthy the

reading is. Another is to widen the control limits by a fixed

safety factor; this is arbitrary and applies the same slack to

precise and imprecise readings alike. A third is to treat every

reading as a fuzzy number and defuzzify before signalling,

which reintroduces a crisp value at the last step and again

discards the ambiguity. None of these separates the two questions

a quality engineer actually cares about: is the process

shifting, and is my measurement good enough to tell.

Neutrosophic statistics, developed by Smarandache and extended

by Aslam and co-workers, generalises classical statis-